Sharp Coefficient Bounds for a New Subclass of q-Starlike Functions Associated with q-Analogue of the Hyperbolic Tangent Function

Let an analytic function $\eta$ in the open unit disk assume to satisfy the conditions $\eta \left(0\right)=0$ and ${\eta }^{{}^{\prime }}\left(0\right)=1,$ and all such types of functions contained in class $\mathcal{A}$ and every $\eta \in \mathcal{A}$ have the following series of the form The set of all such types of univalent functions, which are normalized by the conditions $\eta \left(0\right)=0$ and ${\eta }^{{}^{\prime }}\left(0\right)=1,$ is denoted by $\mathcal{S}.$ A function $\eta$ is called a starlike function if $\eta$ maps U onto the starshape domain. Moreover, the set of all univalent functions which satisfy the condition is denoted by the class ${\mathcal{S}}^{*}$ of starlike functions.

An analytic function w along the following conditions and is called Schwarz function. For two functions, $\eta$ and $g,$ which are analytic in U, we say that $\eta$ is subordinate to g (denoted by $\eta \prec g),$ if there exists a Schwarz function w, such that In particular, if g is univalent in U, then and The class $\mathcal{P}$ represents the class of Caratheodory functions [1], which satisfy the conditions For every $p\in \mathcal{P}$, there is a Taylor series expansion of the form Although function theory was initiated in 1851, in 1916, Bieberbach [2] conjecture unfolded this field and gave a new direction for research. In 1985, De-Branges [3] proved Bieberbach conjecture. A number of outstanding scholars found some new sub-families of the class $\mathcal{S}$ of normalized univalent functions associated with different image domains, for example, starlike $\left({\mathcal{S}}^{*}\right)$ and convex $\left(\mathcal{K}\right)$ functions, respectively. These classes are primary and remarkable subclasses of the univalent class $\mathcal{S}.$ In 1992, Ma and Minda [4] made a very good contribution and defined the general form of the family of univalent functions as follows: where $\phi$ is an analytic function along the conditions $\phi \left(0\right)>0$ and $\mathrm{Re}\left(\phi \right(z\left)\right)>0$ in U. If we choose $\phi \left(z\right)=\frac{1+z}{1-z}$ in (4), then we have a class of starlike functions which is given below:Janowski [5] investigated the class of Janowski starlike functions and defined it as follows:Sokól and Stankiewicz [6] chose $\phi \left(z\right)=\sqrt{1+z}$ and defined the family of class ${\mathcal{S}}_{\mathcal{L}}^{*}$ as The function $\phi \left(z\right)=\sqrt{1+z}$ maps the region U onto the image domain, which is bounded by Recently, Cho et al. [7] chose $\phi \left(z\right)=1+sinz$ and defined a class $\left({\mathcal{S}}_{sin}^{*}\right)$ of starlike functions:Recently, Bano and Raza [8] chose $\phi \left(z\right)=cosz$, and Alotaibi et al. [9] considered $\phi \left(z\right)=coshz$; they defined a family of new classes as follows: and Kumar and Arora [10] considered a function $\phi \left(z\right)=1+{sinh}^{-1}z$ and used a petal-shaped domain and defined a class of starlike functions:In 2021, Barukab et al. [11] investigated a class ${\mathcal{R}}_s$ and found a third Hankel determinant.

For $\eta \in \mathcal{A},$ the jth Hankel determinant is defined by where $n,j\in \mathbb{N},$ and $a_1=1.$

For different values of j and n, the ${\mathcal{H}}_{j,n}\left(\eta \right)$ has the following form:

In 1966, Pommerenke [14] investigated the Hankel determinants for univalent starlike functions. In 1983, Noor [15] defined a class of close-to-convex functions of higher order and investigated Hankel determinants while Ehrenborg [16] investigated the Hankel determinants associated with exponential function.

Recently, numerous researchers have started to pay more attention to finding the sharp bounds of Hankel determinants for a particular family of functions. For example, Janteng et al. [13] calculated the sharp bounds of the second Hankel determinant of the subfamily $(\mathcal{K},{\mathcal{S}}^{*},\mathcal{R})$ of univalent functions class $\mathcal{S}.$ Cho et al. [17] investigated classes of starlike functions of order $\beta$ and strongly starlike functions of order $\beta$, and established a Hankel determinant and showed that $\left|{\mathcal{H}}_{2,2}\left(\eta \right)\right|$ is bounded by ${\left(1-\beta \right)}^2$ and ${\beta }^2.$ Zaprawa [18] used the new methodology and investigated the third Hankal determinant for the class of starlike $\left({\mathcal{S}}^{*}\right)$ and convex $\left(\mathcal{K}\right)$ functions:In 2018, Kwon et al. [19] improved the Zaprawa’s result and proved that Again in 2021, Zaprawa et al. [20] improved the result of Kwon et al. and proved that Jackson [21] used the idea of basic q-calculus and defined the q-analogues of derivatives $\left(D_q\right)$, and further, Ismail et al. [22] used this operator and defined a q-analogous of starlike functions. Recently, number of researchers started to use the $D_q$ and defined many new subclasses of starlike and convex functions and found sharp bounds of second- and third-order Hankal determinants. For example, in 2019, Mahmood et al. [23] defined the family of q-starlike functions and investigated the third-order Hankel determinant, and for close-to-convex functions the same work was carried out by Srivastava in [24]. Arif et al. [25] used the technique of subordination and defined a new subclass of starlike functions associated with sine function [26] and then determined the third Hankel determinant for this class. Meanwhile, Srivastava et al. [27] defined another new subclass of q-starlike functions and investigated Hankel and Toeplitz determinants related with the generalized conic domain. In 2022, Raza et al. [28] defined a new class of q-starlike functions associated with symmetric Booth Lemniscate and studied Hankel determinants. For further research, see [29,30].

The calculus without limits is called q-calculus and it can be dealt with just like classical calculus. However, in q-calculus we do not use limits. Here, we define the q-derivative operator of a function $\eta \in \mathcal{A}$.

By using the definition of subordination, we introduce a new class ${\mathcal{S}}_s^{*}(l,q)$ of q-starlike functions associated with the Sălăgean q-differential operator and connected with the q-analogue of the hyperbolic tangent function.

We need the following lemmas to investigate the sharp coefficient problems for the class ${\mathcal{S}}_s^{*}(l,q).$

Bound of $\left|H_{3,1}\left(\eta \right)\right|$ for the class ${\mathcal{S}}_s^{*}(l,q).$

Considering $l=1,$ in Theorem 2, we obtain the result which is proved in [33].

Let $l=1$ and $q\to 1-,$ in Theorem 1, we obtain the following corollary proved in [37]:

Let $l=1$ and $q\to 1-$ in Theorem 1, we obtain the following corollary proved in [30]:

In 1960, Zalcman defined the conjecture for univalent functions. He stated that every $\eta \in \mathcal{S}$ of the form (1) satisfies the following inequality:In 1999, Ma [38] proved a generalized version of Zalcman conjecture and stated that every univalent function $\eta \in \mathcal{S}$ satisfies the following inequality:

In [39,40], authors discussed the Fekete–Szego functional, the second-order Hankel determinant, and Zalcman conjecture, and these results are shown to be sharp. Furthermore, we have estimated the bounds of the third-order Hankel determinant for this class ${\mathcal{S}}_s^{*}(l,q)$ for different values of n and i. For $n=2$, the inequality (33) has the form

Considering $l=1$ in Theorem 2, we obtain the result which is proved in [33].

Considering $l=1$ and $q\to 1-$ in Theorem 2, we have the following known corollary proved in [37]:

Take $n=3$, $i=2$, in the inequality (34), then we have $\left|a_4-a_2{a}_3\right|\le 2$. Now we discuss it as follows:

Considering $l=1$ in Theorem 3, we obtain the following known corollary, proved in [33]:

Considering $l=1$ and $q\to 1-$ in Theorem 3, we obtain the following known corollary, proved in [37]:

In the following result, we prove the second Hankel determinant $H_{2,2}\left(\eta \right).$